# Colour of your HAT ???

anuragsinghchauhan
Posers and Puzzles 15 Jun '07 05:35
1. 15 Jun '07 05:35
A king wants his daughter to marry the smartest of 3 extremely intelligent young princes, and so the king's wise men devised an intelligence test.

The princes are gathered into a room and seated, facing one another, and are shown 2 black hats and 3 white hats. They are blindfolded, and 1 hat is placed on each of their heads, with the remaining hats hidden in a different room.

The king tells them that the first prince to deduce the color of his hat without removing it or looking at it will marry his daughter. A wrong guess will mean death. The blindfolds are then removed.

You are one of the princes. You see 2 white hats on the other prince's heads. After some time you realize that the other prince's are unable to deduce the color of their hat, or are unwilling to guess. What color is your hat?

Note: You know that your competitors are very intelligent and want nothing more than to marry the princess. You also know that the king is a man of his word, and he has said that the test is a fair test of intelligence and bravery.
2. 15 Jun '07 15:16
"my hat is black"
3. 17 Jun '07 05:49
Originally posted by anuragsinghchauhan
A king wants his daughter to marry the smartest of 3 extremely intelligent young princes, and so the king's wise men devised an intelligence test.

The princes are gathered into a room and seated, facing one another, and are shown 2 black hats and 3 white hats. They are blindfolded, and 1 hat is placed on each of their heads, with the remaining ...[text shortened]... a man of his word, and he has said that the test is a fair test of intelligence and bravery.
my hat is white
4. 17 Jun '07 06:43
I don't think it's possible to be certain one way or the other:

Either of the other princes sees a white hat and mine. If mine is white, then he sees 2 white hats and his could be white or black, so he says nothing. I mine is black, then he sees a white hat and a black hat and his could be white or black, so he says nothing.

I would see 2 white hats and so mine could be either white or black.

I suppose I could play the odds and say "mine is black", in hopes that the hats were distributed randomly, so I would have a 2/3 chance of being correct.
5. 17 Jun '07 07:05
He rewards both intelligence and bravery.
You're being brave by saying something without complete knowledge
You're being intelligent by the fact you have 66% chance of having a black hat

That was how I reached my conclusion
6. 17 Jun '07 11:19
Originally posted by smomofo
I don't think it's possible to be certain one way or the other:

Either of the other princes sees a white hat and mine. If mine is white, then he sees 2 white hats and his could be white or black, so he says nothing. I mine is black, then he sees a white hat and a black hat and his could be white or black, so he says nothing.

I would see 2 white hats ...[text shortened]... opes that the hats were distributed randomly, so I would have a 2/3 chance of being correct.
Let's say your hat is black and the other two are white. This means that each of the other men sees one black hat and one white hat. If there were two black hats, someone would see them and thus know immediately that his hat was white. Since this is obviously not the case, both people seeing the black hat should, after a minute or two, figure out that their hats must be white. Since nobody seems to have done this after a few minutes, the only logical conclusion is that all of the hats must be white. Of course, the others could be faking you out, but he did make a point that they want to win as much as you, so why would they not guess if they knew the answer? Also, this seems the only way to make it completely fair to all of the players, as with any other colour combo one player would have an unfair advantage.
7. 17 Jun '07 14:03
Originally posted by whiterose
Let's say your hat is black and the other two are white. This means that each of the other men sees one black hat and one white hat. If there were two black hats, someone would see them and thus know immediately that his hat was white. Since this is obviously not the case, both people seeing the black hat should, after a minute or two, figure out that thei ...[text shortened]... o all of the players, as with any other colour combo one player would have an unfair advantage.
But if one of the others saw two white hats, they would be able to reason similarly.
8. 17 Jun '07 19:34
Originally posted by anuragsinghchauhan
A king wants his daughter to marry the smartest of 3 extremely intelligent young princes, and so the king's wise men devised an intelligence test.

The princes are gathered into a room and seated, facing one another, and are shown 2 black hats and 3 white hats. They are blindfolded, and 1 hat is placed on each of their heads, with the remaining ...[text shortened]... a man of his word, and he has said that the test is a fair test of intelligence and bravery.
It's been over ten years since I looked at a probability text book so I may have analysed this incorrectly.

Let's look at the case where the three princes all have white hats. The chances of this event happening are 27/125 or 3/5*3/5*3/5 so there is almost an 80% chance that one of them will have a different coloured hat than the others. This is important to remember.

Our prince sees two white hats so ostensibly there should be a 1/3 chance his hat is white and a 2/3 chance his hat is black. If another prince sees a black hat and a white hat the chances that his hat is black are 1/3 and the chances of his hat being white are 2/3. So it seems that there is no difference but we have already shown that the chances of there being 3 white hats is 27/125 so our prince, in this situation, does not have the odds 2/3 but the even more favourable odds of 98/125, almost 80%.

The added dynamic in this case is that quite some time passes and no-one guesses. This can make our prince feel even more confident that he must be wearing a different coloured hat to the other two. Since if the other two could see two white hats, and were smart enough to realise that there is an 80% chance that their hat is black (let's not forget they're brave too!) then one of them, at least, would already have guessed.

So whilst not definitive I think our prince would guess that his hat is black given the 80%(almost) probability and given the unwillingness of the other two to venture a guess.
9. 17 Jun '07 20:18
Originally posted by demonseed
It's been over ten years since I looked at a probability text book so I may have analysed this incorrectly.

Let's look at the case where the three princes all have white hats. The chances of this event happening are 27/125 or 3/5*3/5*3/5 so there is almost an 80% chance that one of them will have a different coloured hat than the others. This is importa ...[text shortened]... the 80%(almost) probability and given the unwillingness of the other two to venture a guess.
However, these probabilities would only work if the king placed the hats on the princes' heads at random, of which there is no indication. As I said before, the most likely probability is that the king gave each of the princes the same coloured hat, thus giving them all an equal chance of winning.
10. 17 Jun '07 20:19
Originally posted by doodinthemood
But if one of the others saw two white hats, they would be able to reason similarly.
Exactly. The king wanted to find out who would figure it out first, and give that person the princess.
11. 17 Jun '07 20:241 edit
Originally posted by whiterose
However, these probabilities would only work if the king placed the hats on the princes' heads at random, of which there is no indication. As I said before, the most likely probability is that the king gave each of the princes the same coloured hat, thus giving them all an equal chance of winning.
There is an indication:

You also know that the king is a man of his word, and he has said that the test is a fair test of intelligence and bravery.

If it's an intelligence problem then it's a logic problem and it works quite well as a logic problem as I have described in the previous post. Since if the king randomly assigns each a hat we can come to a reasonable conclusion that is 80% sure in probability terms but way over 90% because none of the other princes have volunteered an answer.

Your use of the word probability is not in the mathematical sense and when it comes down to it it's rather obvious that this is a logic problem and not a problem about assessing the King's predilection for whimsical fancy.
12. 17 Jun '07 21:17
Originally posted by demonseed
There is an indication:

[b]You also know that the king is a man of his word, and he has said that the test is a fair test of intelligence and bravery.

If it's an intelligence problem then it's a logic problem and it works quite well as a logic problem as I have described in the previous post. Since if the king randomly assigns each a hat we can c ...[text shortened]... logic problem and not a problem about assessing the King's predilection for whimsical fancy.[/b]
The fact that the king says it is a fair test of intelligence almost instantly says to me that they must all have the same colour hat on. Any other situation would result in 1 or 2 of the men having an unfair advantage over the others.

If there were 2 black hats and 1 white then the person with the white hat would instantly know that his was white, making it unfair. If there were 2 white hats and 1 black hat, then the person with the black hat would be at a disadvantage to the other 2, since the other 2 men, after seeing 1 white hat and 1 black hat, would know that if the man wearing a white hat could see 2 black hats then he would instantly know his hat must be white. Therefore they could deduce that their hats must be white.

The only distribution of hats that makes it a fair test of intelligence and bravery is giving everyone a white hat.

Note that while placing hats randomly would give each prince an equal chance of marrying the princess, it would not give each of them an equal chance of proving their intelligence and bravery.
13. 17 Jun '07 22:221 edit
Originally posted by whiterose
Let's say your hat is black and the other two are white. This means that each of the other men sees one black hat and one white hat. If there were two black hats, someone would see them and thus know immediately that his hat was white. Since this is obviously not the case, both people seeing the black hat should, after a minute or two, figure out that thei ...[text shortened]... o all of the players, as with any other colour combo one player would have an unfair advantage.
Whiterose gave the correct answer. It's a pure logic problem, and not a problem of probability.

a) If someone saw 2 black hats, then they'd know for sure that their hat was white.

b) If someone saw 1 black hat. They'd be able to work out that their hat was white, because nobody used the logic in part a

c) If someone saw 0 black hats (2 white hats), then you know that no-one sees any black hats since they haven't stated their position using the logic in a and b.

Since the other two opponents are also wise, and they have the required motivation, you know they're not going to lie to you.
14. 17 Jun '07 22:24
There are two tacts to take here..

The first is the fairness doctrine, which would indicate all hats are the same color, lest the different prince be at an advantage or disadvantage.

The second is as follows (similar to other threads of this sort):

If either of the other princes saw may hat was black, they would then realize theirs was white, or else the other prince in white would see two blacks and immediately know his was white. Clearly they done, and so mine is white as well.
15. 18 Jun '07 01:54
Originally posted by Irax
Whiterose gave the correct answer. It's a pure logic problem, and not a problem of probability.

a) If someone saw 2 black hats, then they'd know for sure that their hat was white.

b) If someone saw 1 black hat. They'd be able to work out that their hat was white, because nobody used the logic in part a

c) If someone saw 0 black hats (2 white hats), th ...[text shortened]... also wise, and they have the required motivation, you know they're not going to lie to you.
I called it a logic problem but you are quite right I treated it like a probability problem. I wasn't too convinced by whiterose's explanation but yours removes any shadow of a doubt. I like those ones particularly when they produce an answer that is counter-intuitive.

**Sigh** Yet more sucky, mechanical, hacneyed thinking: I appear to be going through the motions recently.

Nice answer though, thanks Irax, and whiterose.